Science:Math Exam Resources/Courses/MATH152/April 2012/Question 08 (c)/Solution 1

The complex solution $\vec{𝐳} (t)$ that we found in (a) and (b) is

\begin{aligned} \vec{𝐳} (t) & = e^{- t} e^{2 t i} (\begin{matrix} 1 \\ - i \end{matrix}) \\ = e^{- t} (\cos 2 t + i \sin 2 t) (\begin{matrix} 1 \\ - i \end{matrix}) \\ = e^{- t} (\begin{matrix} \cos 2 t \\ \sin 2 t \end{matrix}) + i e^{- t} (\begin{matrix} \sin 2 t \\ - \cos 2 t \end{matrix}) \end{aligned}

From this we can read off the real-valued form as

\vec{𝐱} (t) = C_{1} e^{- t} (\begin{matrix} \cos 2 t \\ \sin 2 t \end{matrix}) + C_{2} e^{- t} (\begin{matrix} \sin 2 t \\ - \cos 2 t \end{matrix})